Abstract.

The aim of this paper is to obtain quasimodes for a Schrödinger type operator Ph in a semi-classical limit \( (h \searrow 0) \) with exponentially small error terms which are associated with Gevrey families of KAM tori of its principal symbol H. To do this we construct a Gevrey quantum Birkhoff normal form of Ph around the union \( \Lambda \) of the KAM tori starting from a suitable Birkhoff normal form of H around \( \Lambda \). As an application we prove sharp lower bounds for the number of resonances of Ph defined by complex scaling which are exponentially close to the real axis. Applications to the discrete spectrum are also obtained.